Power electronics are becoming increasingly important in modern ship board power systems. They are used for development and implementation of controls and to interface different electrical modules such as loads, batteries and generators that produce, store or consume energy. Power electronic systems are suitable for control because they provide fast operation in the range of microseconds. However, fast control operations can also add complexity to the system. Unintended complex dynamics can arise if the system (inertia) is unable to adapt to the control actions. This article studies the complexity and emergent phenomena that may develop in a ship board power system because of the power electronic components, coupling between these components, as well as feedback mechanisms such as control loops and those of human and environmental origin. A statistical complexity measure, referred to as structural complexity, is used to quantify the degree of complexity that arises during the system evolution. This metric of complexity is computed using permutation entropy and ordinal patterns. A modified procedure for structural complexity called multivariable structural complexity is developed to compute the system wide complexity. This multivariable structural complexity is also used to assist the modelling process and the decision of the best model candidate that captures observed aspects of complexity from the system data history. Various case studies on complexity quantification are conducted on simulated data from a noise coupled buck converter, two parallel connected buck converters and an electric ship board power system.